Vector Spaces and Matrices in Physics

The theory of vector spaces and matrices is an essential part of the mathematical background required by physicists. Most books on the subject, however, do not adequately meet the requirements of physics courses-they tend to be either highly mathematical or too elementary. Books that focus on mathematical theory may render the subject too dry to hold the interest of physics students, while books that are more elementary tend to neglect some topics that are vital in the development of physical theories. In particular, there is often very little discussion of vector spaces, and many books introduce matrices merely as a computational tool.

Vector Spaces and Matrices in Physics fills the gap between the elementary and the heavily mathematical treatments of the subject with an approach and presentation ideal for graduate-level physics students. After building a foundation in vector spaces and matrix algebra, the author takes care to emphasize the role of matrices as representations of linear transformations on vector spaces, a concept of matrix theory that is essential for a proper understanding of quantum mechanics. He includes numerous solved and unsolved problems, and enough hints for the unsolved problems to make the book self-sufficient.

Developed through many years of lecture notes, Vector Spaces and Matrices in Physics was written primarily as a graduate and post-graduate textbook and as a reference for physicists. Its clear presentation and concise but thorough coverage, however, make it useful for engineers, chemists, economists, and anyone who needs a background in matrices for application in other areas.

Table of Contents:
ALGEBRAIC SYSTEMS: AN INTRODUCTION
Abstract Algebraic Systems
Properties of Binary Operations
Group
Field
Ring
Functions or Mappings
VECTOR SPACES
Generalization from Physical to Abstract Vectors
Vector Space
Subspace
Linear Combination of Vectors
Linear Independence and Dependence
Basis and Dimension: Coordinates
Isomorphism of Vector Spaces
Inner Product of Vectors
Norm (or Length) of a Vector
Distance Between Two Vectors
Schwarz Inequality
Orthogonality
LINEAR TRANSFORMATIONS
Definition
Equality
Suma and Scalar Multiple
Zero Transformation
Idempotent Transformation
Nilpotent Transformation
Nonsingular Transformation
Orthogonal Transformation
BASIC MATRIX ALGEBRA AND SPECIAL MATRICES
Definition
Equality
Sum and Difference of Matrices
Scalar Multiple of a Matrix
Matrix Multiplication
Row and Column Vectors
Transpose of a Matrix
Conjugate of a Matrix
Conjugate-Transpose (Hermitian-Conjugate of a Matrix)
Trace of a Square Matrix
Special Square Matrices
Adjoint of a Matrix
Determination of Inverse of a Matrix
Vector Space of Matrices
RANK OF A MATRIX
Row and Column Vectors of a Matrix
Rank of a Matrix
Row Space and Column Space of a Matrix
Elementary Row Operations on a Matrix
SYSTEMS OF LINEAR EQUATIONS
Homogeneous and Non-Homogenous Linear Systems
Matrix Form of a Linear System
Existence and Uniqueness Theorems
A Practical Method of Solving Linear Systems: Gauss Elimination
MATRICES AND LINEAR TRANSFORMATIONS
Matrix Representation of a Linear Transformation
Representation of Product of Transformations
Change of Bases and Similarity Transformation
EIGENVALUES AND EIGENVECTORS OF A MATRIX
Eigenvalues and Eigenvectors
Determination of Eigenvalues and Eigenvectors
Linear Independence of Eigenvectors
Eigenvalues and Eigenvectors of Similar Matrices
Eigenvalues of a Diagonal Matrix
Hermitian, Skew-Hermitian, and Unitary Matrices
Diagonalization of a Matrix
Simultaneous Diagonalization and Commutativity
An Application: Reduction of Coupled Differential Equations of Matrix Eigenvalue Problem
CALEY-HAMILTON THEOREM. MINIMAL POLYNOMIAL OF A MATRIX
Caley-Hamilton Theorem
Determination of Inverse of a Matrix
Minimal Polynomial of a Matrix
A Criterion for Diagonalizability
FUNCTIONS OF A MATRIX
Power of a Matrix
Matrix Polynomial
Matrix Power Series
Evaluation of Matrix Functions for Diagonalizable Matrices
Evaluation of Matrix Functions Using Minimal Polynomial
BILINEAR, QUADRATIC, HERMITIAN, AND SKEW-HERMITIAN FORMS
Bilinear Form
Quadratic Form
Hermitian (Skew-Hermitian) Form
Reduction of a Quadratic Form to Canonical Form
Principal Axes Transformation
ANSWERS/HINTS TO EXERCISES